Separability manifest

Electrophoretic methods are widely used alternatives for the analytical determination of the enantiomeric purity of chiral compounds [194]. Due to the high elTi-ciency of capillary electrophoresis, separations can be achieved even when very low selectivities are observed. At a preparative scale, these methods are well established for the purification of proteins and cells [195] but there is very little published on enantioselective separations. Only recently, some interest in chiral preparative applications has been manifested. Separation of the enantiomers ofterbu-taline [196] and piperoxan [197] have been reported by classical gel electrophoresis using sulfated cyclodextrin as a chiral additive, while the separation of the enantiomers of methadone could be successfully achieved by using free-fluid isotachophoresis [198] and by applying a process called interval-flow electrophoresis [199]. 

The discussion of the exponential ansatz leads us to the concept of manifest separability. Let the fragment wave functions in some model be denoted by t AfPA) and b(Pb). where pa and pb represent the variational parameters. The compound wave function ab(Pa. Pb, Pab) must contain parameters that refer to each fragment A and B separately as well as parameters pab that refer to both fragments simultaneously. The compound wave function and more generally the model itself are said to be manifestly separable if the following requirement is fulfilled for all values of Pa and Pb... 

Thus, for a manifestly separable compound wave function, the interaction parameters Pab become 0 and the wave function turns into a product wave function as the interactions between A and B vanish. Manifestly separable wave functions do not require the use of interaction parameters Pab to describe noninteracting systems size-extensively and the number of variational parameters therefore scales linearly with the number of noninteracting. systems. Obviously, the exponential form of the wave function is manifestly separable, whereas the linear expansion suffers from the lack of this property. 

The Cl approach to the many-electron problem suffers from two main disadvantages. First, since the Cl model is not manifestly separable, it does not provide size-extensive energies. In Section 4.3.2, we found that size-extensivity follows only when the variational space of the compound system is equal to the direct product of the fragment spaces. This requirement cannot be satisfied for Cl wave functions truncated at a fixed excitation level relative to the reference configuration. For example, if two fragments are each described at the CISD level, then a size-extensive treatment of the compound system requires the wave function to be augmented with certain triples and quadruples - namely, those that represent products of single and double excitations in the two subsystems. Such wave functions are not defined within the hierarchy of truncated Cl wave functions based on excitation levels. 

As noted in Section 5.6, the Cl hierarchy of wave functions suffers from two serious shortcomings the lack of size-extensivity and the slow convergence towards the FCI limit. We shall now see how these problems are overcome through the introduction of the coupled-cluster model, which represents a nonlinear but manifestly separable parametrization of the correlated electronic state. The coupled-cluster model constitutes a particularly successful approach to electronic-structure theory, providing for many purposes - often in combination with perturbation theory - the most efficient strategy for the accurate calculation of electronic energies and wave functions. 

This coupled-cluster wave function is manifestly separable (see Section 4.3.4) and differs from the Cl wave function (5.6.5) by the presence of terms that are nonlinear in the excitation operators. Since the excitation operators such as (5.6.6) and (5.6.7) commute with one another, there is no problem with the wder of the operators in the product wave function (5.7.1). 

When ethanoi is present, the risk of separation is much less than with methanol. Nevertheless, the ethanol should be relatively anhydrous (less than 3000 ppm water) moreover, if a fuel containing ethanol comes in contact with a water layer, a migration of ethanol toward the water is observed creating a fuel quality problem manifested by lower octane number and an environmental quality problem in that the water will need to be treated. Distribution of ethanol-based fuels requires extra precaution to ensure dryness in distribution systems. 

The native form of chromatin in cells assumes a higher order stmcture called the 30-nm filament, which adopts a solenoidal stmcture where the 10-nm filament is arranged in a left-handed cod (Fig. 5). The negative supercoiling of the DNA is manifested by writhing the hehcal axis around the nucleosomes. Chromatin stmcture is an example of toroidal winding whereas eukaryotic chromosomes are linear, the chromatin stmctures, attached to a nuclear matrix, define separate closed-circular topological domains. 

Filter aids may be applied in one of two ways. The first method involves the use of a precoat filter aid, which can be applied as a thin layer over the filter before the suspension is pumped to the apparatus. A precoat prevents fine suspension particles from becoming so entangled in the filter medium that its resistance becomes exces-sive. In addition it facilitates the removal of filter cake at the end of the filtration cycle. The second application method involves incorporation of a certain amount of the material with the suspension before introducing it to the filter. The addition of filter aids increases the porosity of the sludge, decreases its compressibility, and reduces the resistance of the cake. In some cases the filter aid displays an adsorption action, which results in particle separation of sizes down to 0.1 /i. The adsorption ability of certain filter aids, such as bleached earth and activated charcoals, is manifest by a decoloring of the suspension s liquid phase. This practice is widely used for treating fats and oils. The properties of these additives are determined by the characteristics... 

The problem manifested itself as an equipment problem, namely a leaking flange joint. The department broadly responsible for this area (but not for implementing, monitoring, and subsequent recommendations) is the engineering department, as the specialist contractors work for them. The critical event took place during a corrective maintenance operation. From here, two separate root causes were identified, based on the data from the investigation. 

This novel manifestation of the gas chromatographic separation demands that Our fundamental understanding of the GC method-invariably of single-dimensional scope-is challenged as follows concepts of column efficiency and separation are now Supplanted by a need to compare the performances of two columns operating... 

Flame dynamics is intimately related to combustion instability and noise radiation. In this chapter, relationships between these different processes are described by making use of systematic experiments in which laminar flames respond to incident perturbations. The response to incoming disturbances is examined and expressions of the radiated pressure are compared with the measurements of heat release rate in the flame. The data indicate that flame dynamics determines the radiation of sound from flames. Links between combustion noise and combustion instabilities are drawn on this basis. These two aspects, usually treated separately, appear as manifestations of the same dynamical process. 

Hence, in order to obtain a successful separation of a mixture of the H-bonded analytes, the most recommended are such chromatographic systems in which lateral interactions either are entirely eliminated or manifest themselves to a rather weak extent. Thus, it is evident that in such cases, the most recommended are the chromatographic systems composed of an active enough adsorbent and a polar enough mobile phase. 

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